Experimental study on a compressible vortex ring in collision with a wall

Abstract

Head-on collision of the compressible vortex ring with a wall is studied experimentally for shock Mach number (M) varying from 1.31 to 1.85 using high-speed smoke flow visualizations. The compressible vortex ring is generated from the open end of a short driver section shock tube. At first, the interaction of the isolated compressible vortex ring with wall is characterized using diameter and translational velocity of the vortex ring and verified with flow visualizations. Different events such as vortex ring impingement on wall, wall vortex growth, and lift-off are identified from the change in slope of diameter and translational velocity. Next, the wall interaction of primary vortex ring embedded with counter rotating vortex ring (CRVR) is studied for M = 1.7 and 1.85. It has been observed that the presence of CRVR strongly influences the flow structure and formation of wall vortices. A case where CRVR first impinges on the wall showed a lesser lift-off due to formation of weak wall vortices. Strong scattered waves produced during wall interaction at high M (1.7 and 1.85) interact with trailing jet and create a secondary vortex ring at the trailing jet. Pressure measurements along the wall surface have also showed the impingement, wall vortex growth, lift-off besides shocklet formation.

Graphical Abstract

Graphical Abstract text

1 Introduction

Studies on compressible vortices interacting with a wall is essential in understanding the ground effects of flying machines, acoustic loads on launch vehicles, design of launch pads, thrust vectoring and enhancement of cooling in manufacturing processes. There are also other engineering flows such as flow past buildings, suspension cables and cooling towers which need the understanding of turbulence, formation and interaction vortices to minimize the drag. Vortex ring is a simple vortical structure with concentrated vorticity. It can be generated in the laboratory by ejecting slug of fluid through a nozzle, orifice and the open end of a shock tube. Wall interaction of the incompressible vortex ring was studied by many researchers (Walker et al. 1987; Lim 1989; Lim et al. 1991; Naitoh et al. 2001; Duport et al. 2002). Unlike incompressible ring, there is a significant change in flow and fluid properties across the compressible vortex ring. Formation of shocklets and their interaction with vortices increases further the complexity at high Mach numbers (M). Minota et al. (1997) initiated the compressible vortex ring–wall interaction studies using shadowgraph and numerical simulations for M = 1.35 by placing a plate at x/D = 2.68. They observed two shocklets, one between the primary and wall vortex and another shocklet between the primary vortex core and the separation point of the boundary layer. They proposed that the shocklet formation is similar to the shock formation mechanism in convergent–divergent nozzle flows.

Szumowski et al. (2000) studied the starting jet–wall interaction (x/D = 2) using the schlieren technique for M varying from 0.65 to 1.14. They have noticed the generation of toroidal waves during shock–vortex ring interaction and measured the wall pressure along the radial direction. Kontis et al. (2006) studied the compressible vortex ring interacting with a wall at x/D = 3.33 using schlieren images for M = 1.28, 1.52 and 1.63. It was observed that the initial shock strength plays a dominant role in wall interaction and a spherical shock was produced at wall at higher M. In another study, Kontis et al. (2008) reported a region of strong flow development near the wall and generation of multiple pressure fluctuations in the near field. Studies on vortex ring interacting with a wall using schlieren and shadowgraph techniques show the discontinuities in density such as shock and scattered waves. However, the flow field information along the depth is lost as it is embedded within the image due to the integral nature of the visualization technique (planar projection of 3D flow). Sectional views of vortex ring give a better understanding as the flow field is highly three-dimensional at high Mach numbers. They are obtained by feeding smoke in the flow and passing a light sheet at any plane to illuminate the smoke particles. The nature of the vortex ring (laminar or turbulent) and the formation of trailing jet and wall vortices are easily obtained using smoke flow visualizations.

In earlier studies, the wall was kept close to shock tube exit (Minota et al.: x/D = 2.68, Szumowski et al.: x/D = 2, Kontis et al.: x/D = 3.33). Hence the vortex ring was followed by a strong trailing jet. The interaction of an isolated compressible vortex ring with a wall is lacking in the literature. Various phenomena of wall interaction such as impingement, boundary layer separation, wall vortex growth, and lift-off are identified by many researchers for incompressible vortex ring (Lim 1989; Lim et al. 1991; Naitoh et al. 2001; Duport et al. 2002; Naguib and Koochesfahani 2004). However, these phenomena are not identified in the compressible vortex ring. In a preliminary study, Murugan et al. (2008) produced the isolated compressible vortex ring using a short driver section length (L = 165 mm) shock tube and observed its wall interaction at x/D = 4.7. They compared the vortex ring impingement and lift-off using flow visualization for M = 1.31 and 1.55.

In the present study, wall interaction of isolated vortex ring and the vortex ring embedded with counter rotating vortex ring (CRVR) are studied in detail using high-speed smoke flow visualizations. Various events of wall interaction are identified from the variation of vortex ring’s diameter and translational velocity measured from flow visualizations. As the vortex ring is strongly influenced by the supersonic trailing jet for M ≥ 1.6 (formation of embedded shock and CRVR), its interaction with a wall is studied separately. The CRVR has a tendency to roll around the primary vortex ring and interacts with the trailing jet (Murugan and Das 2010). Its vorticity is opposite to the primary vortex ring (Murugan and De 2011) which is similar to primary (positive) and wall vortex (negative) vorticities observed during vortex ring–wall interaction. Presence of CRVR with negative vorticity in front of the primary vortex ring near the wall causes the formation of wall vortex with positive vorticity. This special case is also studied here which is not available in the literature. Finally, the fluctuating pressure at wall along the radial direction is measured for identifying the wall vortices and shocklet.

2 Experimental setup and instrumentation

Figure 1 shows the shock tube and various instruments used for flow visualizations. Shock tube maximum driver section length is 315 mm and it is varied by placing cylindrical blocks inside the driver section. Helium is used as a driver section gas as it gives higher shock strength and M compared to air (Murugan 2008). Shock tube-driven section length is 1,200 mm. The inner and outer diameters of the shock tube are 64 and 100 mm, respectively. For studying the isolated vortex ring–wall interaction, an aluminum square plate (450 mm × 450 mm) is kept at x/D = 4.7 from the shock tube exit. For higher M, the CRVR rolls around the primary vortex ring if the plate is placed at x/D = 4.7. Hence the plate is moved to x/D = 3.125 for M = 1.7 and 1.85 to study the effect of CRVR. Shock speed is calculated using signals obtained from two PCB piezoelectric pressure transducers kept inside the shock tube at 300 mm apart and shock M is calculated with respect to ambient condition.

Fig. 1

Experimental setup and apparatus

Shock tube-driven section and the region near the wall are filled with paraffin oil smoke before rupturing the diaphragm for visualizing the vortices. Laser sheet produced from a 532 nm and 10 Hz double pulsed Nd–YAG laser is used for illuminating the smoke flow field. Time delay between the pulses can be varied from 1 μs to 100 ms in the laser panel. A single double shutter imaging CCD camera (PCO, Pixelfly) with 1,024 × 1,024 pixels resolution and 16 Hz frame rate is used for capturing the flow field. Time delay between double shutter imaging is controlled using the camware software. Since the entire evolution of wall interaction is completed in a few milliseconds and the maximum frequency of the laser is 10 Hz, only two images with custom time delay is obtained in every experiment (Murugan and Das 2010). Both laser and camera are synchronized using an external delay generator which is triggered using the signal from a pressure transducer kept inside the shock tube.

Many experiments are performed to get the entire progression of vortex ring–wall interaction for given M. The shock M strongly depends on the area of diaphragm rupture, ambient condition, driver section length and pressure (Murugan et al. 2011a). For a given driver section length and pressure, the characteristics of vortex ring depend on the diaphragm rupture area. If the diaphragm ruptures uniformly (circular) in every experiment, then the variation in translational velocity and diameter is less. For 60 psi driver section pressure, uniform bursting gives the variation of shock speed from 483 to 498 m/s and on an average 5 out of 10 experiments gives the shock speed of 491 m/s. Improper bursting (tearing) of diaphragm gives shock speed as low as 468 m/s. This results in significant change in translational velocity and diameter of the vortex ring. The maximum percentage variation of M (considering uniform bursting) considered in the present study for M = 1.31, 1.45, 1.55, 1.7 and 1.85 are 3, 3.7, 3.6, 4, and 2.8 %, respectively. The plate is opened in the atmosphere to avoid interaction of any reflected shock with the flow field (Fig. 1). Capturing the wall vortices is a painful task due to diffusion of smoke in the ambient air. Since the entire process is controlled manually, any delay in bursting the diaphragm, seeding smoke inside the shock tube and near wall results in blurred images. Furthermore, it is also difficult to distinguish the wall vortex (fully covered with smoke) as clear as primary ring (entrain ambient fluid inside the core).

3 Results and discussion

Complete evolution of wall interaction for any M shown in the present study is obtained by combining images captured from many experiments at different times. Vortex ring diameter is calculated manually from the distance between vortex cores in cross-sectional view of the vortex ring (Fig. 2c). Its translational velocity is calculated using the pixel displacement of vortex cores obtained at 40 μs time difference from double shutter imaging in an experiment. Pixel displacement in the x direction is used for calculating translational velocity along the axial direction. For calculating velocity in the radial direction, pixel displacement along the y direction is used. Since the vortex ring translational and lateral velocities are calculated using pixel displacement in the flow visualizations, they may have ±10 % variation. Time at which the incident shock reaches the shock tube exit is taken as t = 0 which is normalized using the mass velocity behind the incident shock (U _b) and the shock tube inner diameter (D). U _b is calculated from the one-dimensional shock tube relations using experimentally measured incident shock M. Diameter and translation velocity of the vortex ring during evolution are normalized using D and U _b. However, these non-dimensional parameters do not characterize the wall interaction properly as the velocity of impingement varies with M and distance (x). Hence, for normalizing the wall interaction, diameter and translational velocity before impingement are used as both become asymptotic after vortex ring pinches off from the trailing jet.

Fig. 2

Vortex ring evolution and its impingement on wall for M = 1.45

Wall interaction of vortex ring is divided into four parts in the present study. Shock tube driver section length (L) is 115 mm for all cases except flow visualizations at M = 1.7 and 1.85 where L = 315 mm as driver section length is essential for generating CRVR (Murugan 2008). Flow visualizations of vortex ring evolution and wall interaction are studied for M = 1.45 in the first part. Effect of wall on diameter and translational velocity is studied next by comparing with free traveling vortex ring. Third, the different events of wall interactions are identified from the variation of diameter and lateral velocity of the vortex ring for M = 1.31, 1.45 and 1.55. Next the effect of CRVR embedded with the primary vortex ring on wall interaction are studied for M = 1.7 and 1.85. Finally, the variation of fluctuating wall pressure along the radial direction is measured to identify the impingement, wall vortex and shocklet.

3.1 Flow visualization of vortex ring–wall interaction

Figure 2 shows the formation, growth and impingement of vortex ring on wall for M = 1.45. Vortex ring’s evolution is classified into three stages namely formation, growth and free traveling stage. Figure 2a shows the initial roll up and the formation of vortex ring at shock tube exit after diffraction of incident shock. Pressure, velocity and density behind the incident shock (U _b) at the shock tube exit are 2.14 bar, 202 m/s, and 2 kg/m³, respectively. Vortex ring grows continuously after formation due to entrainment of trailing jet fluid in Fig. 2b, c. Trailing jet velocity reduces gradually as the expansion waves from the closed end of the driver section reaches the open end. Vortex ring pinches off from the trailing jet when the trailing jet velocity becomes less than the translational velocity of the vortex ring (Fig. 2d). Figure 2e, f shows the free traveling stage of vortex ring where it moves downstream with its self-induced velocity (95 m/s). Figure 2h–j from t = 2,663 to 2,863 μs shows the impingement of vortex ring on the wall.

Dense smoke in the axial region of vortex ring near the wall in Fig. 2h shows accumulation of particles and increase in pressure. Stretching of vortex ring in the lateral direction due to the inertia of the ring is seen at t = 2,863 μs in Fig. 2i. High-pressure fluid in the axial region also plays significant role in vortex stretching by accelerating flow in the radial direction. Boundary layer induced at the wall separates, rolls up and forms the first wall vortex at t = 3,070 μs in Fig. 2j due to the presence of primary vortex near wall. Figure 3 shows the evolution of wall vortex, lift-off and translation of vortex ring along the radial direction for M = 1.45. Wall vortex lifts off from the surface (in Fig. 3c) with primary vortex after attaining the maximum circulation. It rolls around the primary vortex (in Fig. 3e) and becomes a turbulent structure near the wall (Fig. 3f). However, the primary vortex ring preserves its compact form and moves along the radial direction for long distance (Fig. 3f–i).

Fig. 3

Evolution of wall vortex and translation of vortex ring in the lateral direction for M = 1.45

3.2 Effect of wall on diameter and translational velocity

In this section the effect of wall at x/D = 4.7 is studied by comparing the free traveling vortex ring’s diameter and translational velocity for M = 1.45 in Fig. 4. Velocity behind the incident shock (U _b) at exit for M = 1.45 is 206 m/s. Variation of vortex ring’s diameter up to t* ≈ 6.4 is almost same for both cases. Incident shock leaves the shock tube exit at t* = 0, reflects from the wall kept at x/D = 4.7 and interacts with the vortex ring at t* ≈ 3.5. Hot wire anemometry and acoustic measurements have confirmed the interaction of reflected shock with the vortex ring at t* ≈ 3.5 (Murugan 2008). But, no significant increase in diameter is observed during shock–vortex ring interaction in Fig. 4. Though the vortex ring impinges on wall at t* ≈ 8.5 in Fig. 2h, influence of wall is felt at t* ≈ 7.5. The diameter increases gradually as the vortex ring approaches the wall (inset of Fig. 4). It increases rapidly after impingement as it stretches along the lateral direction (Fig. 2i). This increase in diameter from t* ≈ 8.6 is due to acceleration of high-pressure fluid in the axial region. Significant change in slope at t* ≈ 9.6 in Fig. 4 denotes the formation of wall vortex at t = 3,070 μs in Fig. 3. Variation of diameter is uniform during wall vortex growth up to t* = 10.6. Primary vortex lifts off with the wall vortex at t* ≈ 10.8 and no significant change in diameter is observed during this time for M = 1.45. Any change in slope of diameter after t* ≈ 11 may result from the formation secondary and tertiary wall vortex.

Fig. 4

Non-dimensional diameter variation of vortex ring for M = 1.45

Translational velocity of vortex ring along the axial direction for M = 1.45 is shown in Fig. 5. It is less during formation (t U _b/D = 2; Arakeri et al. 2004) due to roll up of vortex sheet. It increases during vortex ring growth up to t* = 4.2 as the trailing jet expands at the shock tube exit. It reaches an asymptotic velocity during free traveling stage of vortex ring after pinch off. There is a significant reduction in velocity at t* = 3.5 when a plate is kept at x/D = 4.7 due to the interaction of reflected shock with the vortex ring. Vortex ring is attached with the trailing jet during this interaction. This ring also reaches an asymptotic velocity (self-induced velocity) after pinching off. However, the translational velocity is lower than the free traveling vortex ring at t* >6. Vortex ring reaches the wall at t* ≈ 8.5 and translational velocity reduces continuously after that due to adverse pressure gradient. At t* ≈ 10, vortex ring accelerates along the radial direction as the fluid in the axial region expands. Since the translational velocity is calculated based on the vortex core center displacement, stretching of vortex ring along the radial direction also increases the translational velocity from t* >10. Once the wall vortex lifts off from the wall, the primary vortex ring stretches and moves away from the wall (Fig. 3c). This causes negative (opposite) translational velocity of the vortex ring at t* ≈ 10.55. The primary vortex moves towards the wall after wall vortex rolls over it which is seen as a positive translational velocity at t* ≈ 10.75. The translational velocity becomes zero from t* ≈ 11 onwards if further significant wall vortices do not form after the primary vortex reaches the vicinity of the wall.

Fig. 5

Non-dimensional translational velocity variation of vortex ring for M = 1.45

3.3 Effect of Mach number on wall interaction

Figure 6 shows the variation of vortex ring diameter for M = 1.31, 1.45 and 1.55. Vortex ring impinges on the wall for three Mach numbers at different times as the translational velocity varies with M. Hence, the time at which the vortex ring impinges on the wall for each M is subtracted from the absolute time (t) to make impingement time t = 0 for comparing the results in Fig. 6. Increase in diameter up to time t ≈ 0.75 after impingement is due to expansion of flow in the radial direction. Slope change at t ≈ 0.75 corresponds to the growth of wall vortex. Lifting off process of the primary vortex with wall vortex causes another change in slope at t ≈ 1.3 for M = 1.55. This slope change depends on the strength of the wall vortex. Constant slope after t ≈ 1.5 for all M shows the vortex ring translation along the radial direction.

Fig. 6

Comparison normalized diameter variation of vortex ring for M = 1.31, 1.45, and 1.55

Figure 7 shows the variation of vortex ring lateral velocity along the wall for M = 1.31, 1.45 and 1.55. Vortex ring accelerates in the lateral direction after impingement from t ≈ 0 to 0.75 due to expansion of fluid in the axial region besides its self-induced velocity. It is severely stretched during impingement at higher M as its strength increases. Lateral velocity reduces from t ≈ 0.75 as wall vortex grows near the surface and reaches to a minimum value during lift-off at t ≈ 1.3. Lift-off strength increases with an increase in the strength of the vortex ring. Hence a large reduction in lateral velocity is observed during lift-off for higher M. Vortex ring accelerates slightly after the wall vortex lifts off as the flow expands between the primary vortex and wall from t >1.5. Formation and growth of a second wall vortex cause the reduction in lateral velocity again from t >1.9.

Fig. 7

Comparison normalized velocity variation of vortex ring for M = 1.31, 1.45, and 1.55

3.4 Vortex ring–wall interaction at high Mach numbers

Formation of counter rotating vortex ring (CRVR) ahead of the primary vortex ring was first observed by Brouillette and Hebert 1997 for M ≥ 1.6. CRVR after formation rolls around the primary vortex ring and interacts with the trailing jet for M = 1.7 (Murugan and Das 2010). Multiple CRVR are formed as either driver section length or pressure ratio increases. Murugan and De (2011) simulated the impulsive flow from the shock tube for M = 1.7 by solving the axisymmetric Navier–Stokes equations using AUSM + scheme and found the mechanism of CRVR formation. The shear layer appearing at the triple point of the Mach disk is responsible for the roll up and formation of CRVR. In this section, interaction of vortex ring embedded with CRVR is studied qualitatively by keeping a plate at x/D = 3.125 from the shock tube exit.

Figure 8 shows the formation and wall interaction of primary and counter rotating vortex ring for M = 1.7 with L = 315 mm. Here, the vortex ring is followed by a long trailing jet compared to the earlier cases as the trailing jet length is a function of driver section length and M (Murugan 2008). Formation and evolution of strong embedded shock in the axial region are seen in Fig. 8b–f which was studied in detail by Murugan and Das (2010). CRVR appearing in Fig. 8b rolls around the primary vortex ring and impinges on the wall at t = 1,477 μs in Fig. 8h. The scattered waves generated during impingement interact with the trailing jet and form a strong secondary vortex ring at the trailing jet (Fig. 8f–l). Formation, evolution and the lift-off of vortices are seen in Fig. 8g–j. Primary vortex moves close to the wall (Fig. 8k) and translates along the radial direction after wall vortex rolls around it. In Fig. 8, the CRVR moves away from the wall before vortex ring impingement for M = 1.7.

Fig. 8

Wall interaction of vortex ring embedded with CRVR for M = 1.7

Wall interaction of CRVR and its effects on the vortices formation can be studied by either increasing the M or placing the wall at short distance. Here, CRVR wall interaction is studied by increasing the M to 1.85 in Fig. 9. CRVR impinges on the wall with primary vortex ring in Fig. 9c–e. Though the primary vortex ring impinges on wall at t = 987 μs, the wall vortex growth and lift-off are delayed and observed around t = 1,337 μs and t = 1,680 μs. This is due to the presence of CRVR ahead of primary ring which causes the formation of weak wall vortex in Fig. 9g compared to Fig. 8h. The lift-off distance d ₂ (Fig. 8j) for M = 1.85 is also less compared to d ₁ (Fig. 9i) for M = 1.7. This happens only when CRVR stays ahead of the primary vortex ring.

Fig. 9

Wall interaction of vortex ring embedded with CRVR for M = 1.85

3.5 Wall pressure measurements

Figure 10 shows the relative locations of pressure transducers at wall with respect to shock tube exit and vortex ring. Pressure history at the wall (p) is acquired at 100 kHz frequency with a Kistler pressure transducer placed at y = 0, 30, 60, 90, 120 and 150 mm. It is normalized using the ambient pressure (p _a = 101,325 N/m²). In every experiment, pressure is measured at only one location and the progression shown here is obtained from many experiments. Figure 11 shows the normalized pressure measured at various radial locations for M = 1.7 with L = 115 mm. Figure 11a shows the time history of pressure variation at y = 0. A sharp rise in pressure at t ≈ 0.7 ms shows the impingement of incident shock on the wall. The incident shock strength decreases from y = 0 to 150 mm as it expands spherically at the shock tube exit (Murugan 2008). A sharp rise in pressure from t ≈ 2 to 2.2 ms at y = 30 mm shows the impingement of vortex ring on the wall. Decrease in pressure from t ≈ 2.2 to 2.4 ms during vortex ring translation results from the expansion of flow along the radial direction. Another rise in pressure at t ≈ 2.4 ms results from the impingement of trailing jet on the wall. The strength of the impingement reduces in y = 30 and 60 mm at t ≈ 2 ms due to the presence of re-circulating region of the vortex ring. A sharp pressure rise at t ≈ 2.2 ms in y = 60 mm results from the formation of shocklet between primary vortex and separation point of the boundary layer (Minota et al. 1997). Wall vortex growth is identified from pressure signal at t ≈ 2.3 ms in y = 90 mm. There is no distinct pressure peak in y = 120 mm as wall vortex lifts off from the surface. A rise in pressure at t ≈ 2.45 ms in y = 150 mm may be due to the formation of second wall vortex. However, the second wall vortex is not observed in smoke flow visualizations as it is difficult to preserve smoke near the wall for long duration.

Fig. 10

Location of pressure transducers with respect to vortex ring

Fig. 11

Variation of fluctuating pressure at wall along the radial direction for M = 1.7

4 Conclusions

Compressible vortex ring colliding with a wall has been studied experimentally for M varying from 1.31 to 1.85 using smoke flow visualizations and pressure measurements. It has been observed that the vortex ring diameter does not change and the translational velocity reduces significantly after shock–vortex ring interaction. Vortex ring impingement, wall vortex growth and lift-off are identified from diameter, translational and lateral velocity of the vortex ring and verified with flow visualizations. Wall interaction of CRVR ahead of primary ring has delayed the formation of wall vortices and reduced the lift-off strength at high M. Wall pressure measurements at various radial locations have also showed the vortex ring impingement, flow expansion, wall vortex growth, and shocklet formation. Although, the various events of wall interaction are identified from a combination of many experiments in the present study, a time-resolved experiment using high-speed camera and laser will yield finer and accurate slope changes in diameter and translational velocity. The formation of second and third wall vortex is very difficult to observe from experiments. An efficient, validated and accurate numerical simulation is essential for studying the later stage of compressible vortex ring–wall interaction.